On the number of union-free families

József Balogh, Adam Zsolt Wagner

Research output: Contribution to journalArticlepeer-review


A family of sets is union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. Kleitman proved that every union-free family has size at most (1+o(1))( n/2 n). Later, Burosch–Demetrovics–Katona–Kleitman–Sapozhenko asked for the number α(n) of such families, and they proved that 2(nn/2)≤α(n)≤222(nn/2)(1+o(1)) They conjectured that the constant 22 can be removed in the exponent of the right-hand side. We prove their conjecture by formulating a new container-type theorem for rooted hypergraphs.

Original languageEnglish (US)
Pages (from-to)431-448
Number of pages18
JournalIsrael Journal of Mathematics
Issue number1
StatePublished - Apr 1 2017

ASJC Scopus subject areas

  • Mathematics(all)


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